Difference between revisions of "Linearized Einstein equations"

Let \[g^{\mu\nu}=\eta^{\mu\nu}+\tilde{h}^{\mu\nu}.\] Then by definition \[\delta^{\mu}_{\nu}=g^{\mu\lambda}g_{\lambda\nu} =(\eta^{\mu\lambda}+\tilde{h}^{\mu\lambda}+O(h^2)) (\eta_{\lambda\nu}+h_{\lambda\nu}+O(h^2)) =\delta^{\mu}_{\nu}+h^{\mu}_{\nu}+\tilde{h}^{\mu}_{\nu}+O(h^2),\] so \[\tilde{h}^{\mu}_{\nu}=-h^{\mu}_{\nu}.\]

All the tensors considered are constructed of derivatives of the metric tensor, and thus are of the first order by $h$. Partial derivative is tensorial in the zeroth order by $h$: $\partial_{\mu}=\nabla_{\mu}+O(h)$. Also, as shown above, if we use $\eta_{\mu\nu}$ to lower and raise indices, the result will differ from the exact expression in the next order by $h$. Therefore in the assumed approximation we can neglect the difference between the partial and covariant derivatives and freely put the metric in and out of action of differential operators. Using the notation \begin{align*} h&\equiv h^{\mu}_{\mu};\\ \partial^{\lambda}& \equiv \eta^{\lambda\mu}\partial_{\mu};\\ \square&\equiv\eta^{\mu\nu}\partial_{\mu}\partial_{\nu} \equiv \partial^{\mu}\partial_{\mu}, \end{align*} where $\square$ is the usual flat D'Alembertian, we get \begin{align*} \Gamma^{\lambda}_{\mu\nu} &=\tfrac12 g^{\lambda\rho} (\partial_{\mu}h_{\rho\nu}+\partial_{\nu}h_{\rho\mu} -\partial_{\rho}h_{\mu\nu})=\\ &=\tfrac12 (\partial_{\mu}h^{\lambda}_{\nu} +\partial_{\nu}h^{\lambda}_{\mu} -\partial^{\lambda}h_{\mu\nu})+O(h^2);\\ {R^{\mu}}_{\nu\rho\sigma} &=\partial_{\rho}\Gamma^{\mu}_{\nu\sigma} -\partial_{\sigma}\Gamma^{\mu}_{\nu\rho}+O(h^2)=\\ &=\tfrac{1}{2}\big( \partial_{\rho}\partial_{\nu}h^{\mu}_{\sigma} +\underline{\partial_{\rho}\partial_{\sigma}h^{\mu}_{\nu}} -\partial_{\rho}\partial^{\mu}h_{\nu\sigma} -\partial_{\sigma}\partial_{\nu}h^{\mu}_{\rho} -\underline{\partial_{\sigma}\partial_{\rho}h^{\mu}_{\nu}} +\partial_{\sigma}\partial^{\mu}h_{\nu\rho}\big)+O(h^2)=\\ &=\tfrac{1}{2}\big( \partial_{\rho}\partial_{\nu}h^{\mu}_{\sigma} -\partial_{\rho}\partial^{\mu}h_{\nu\sigma} -\partial_{\sigma}\partial_{\nu}h^{\mu}_{\rho} +\partial_{\sigma}\partial^{\mu}h_{\nu\rho}\big) +O(h^2);\\ R_{\nu\sigma} &=\tfrac{1}{2}\big( \partial_{\nu}\partial_{\mu}h^{\mu}_{\sigma} -\square h_{\nu\sigma} -\partial_{\sigma}\partial_{\nu}h +\partial_{\sigma}\partial_{\mu}h^{\mu}_{\nu}\big) +O(h^2);\\ R&\equiv \eta^{\nu\sigma}R_{\nu\sigma}= \partial_{\nu}\partial_{\mu}h^{\mu\nu} -\square h +O(h^2);\\ G_{\mu\nu} &\equiv R_{\mu\nu}-\tfrac{1}{2}R\eta_{\mu\nu}=\\ &=\tfrac{1}{2}\big[ \partial_{\mu}\partial_{\lambda}h^{\lambda}_{\nu} +\partial_{\nu}\partial_{\lambda}h^{\lambda}_{\mu} -\partial_{\mu}\partial_{\nu}h -\square h_{\mu\nu} +\eta_{\mu\nu}(\square h - \partial_{\rho}\partial_{\sigma}h^{\rho\sigma}) \big]. \end{align*}

Taking the trace we get \[\bar{h}=h-2h=-h \quad\text{and}\quad h_{\mu\nu}=\bar{h}_{\mu\nu} -\frac{1}{2}\bar{h}\;\eta_{\mu\nu},\] so now it is clear why $\bar{h}_{\mu\nu}$ can be named the "trace-reversed" metric. Then \begin{align*} G_{\mu\nu}&=\tfrac{1}{2}\big[ \partial_{\mu}\partial_{\lambda}\bar{h}^{\lambda}_{\nu} -\underline{\tfrac{1}{2}\partial_{\mu}\partial_{\nu}\bar{h}} +\partial_{\nu}\partial_{\lambda}\bar{h}^{\lambda}_{\mu} -\underline{\tfrac{1}{2}\partial_{\mu}\partial_{\nu}\bar{h}} +\underline{\partial_{\mu}\partial_{\nu}\bar{h}} -\square \bar{h}_{\mu\nu} +\tfrac{1}{2}\eta_{\mu\nu}\square \bar{h} +\\ &\qquad+\eta_{\mu\nu}(-\square \bar{h} -\partial_{\rho}\partial_{\sigma}\bar{h}^{\rho\sigma} +\tfrac{1}{2}\square \bar{h}) \big]=\\ &=\tfrac{1}{2}\big[ \partial_{\mu}\partial_{\lambda}\bar{h}^{\lambda}_{\nu} +\partial_{\nu}\partial_{\lambda}\bar{h}^{\lambda}_{\mu} -\square\bar{h}_{\mu\nu} -\eta_{\mu\nu} \partial_{\rho}\partial_{\sigma}\bar{h}^{\rho\sigma} \big]. \end{align*}